Histories of mathematical practice: reconstruction, genealogy, and the unruly pasts of ruly knowledge
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ORIGINAL ARTICLE
Histories of mathematical practice: reconstruction, genealogy, and the unruly pasts of ruly knowledge Michael J. Barany1 Accepted: 17 June 2020 © The Author(s) 2020
Abstract Histories of mathematical practice account for mathematical knowledge and action by interpreting presently-available evidence as traces of the events, contexts, and relations that make up the past. Interpretations depend on the assumptions one makes about how mathematical knowledge works, insofar as it is knowledge and insofar as it is mathematical. Though the specific rules and their meanings can differ from context to context, mathematics is a kind of ruly knowledge, expected to follow orderly patterns and principles wherever it is found. The contexts and activities of mathematical practice—how that knowledge is made, shared, applied, and understood—are necessarily less ruly, and different practices leave or occlude different kinds of evidence for historical interpretation. The apparent ruliness of mathematics can be both a resource and an obstacle for understanding its unruly pasts. Historians’ interpretive assumptions and goals have been shaped by centuries of interaction between mathematics research, history, and education. As a guide for mathematics educators and education researchers to historical perspectives on mathematical practice, this article briefly introduces four major interpretive traditions that inform the present discipline of mathematics history. It then illustrates some interpretive approaches and challenges through the history of blackboards in mathematical practice before explaining the two broad kinds of historical interpretation applied to mathematical practices. Reconstruction involves understanding the conditions and contexts of practices in a single historical moment. Genealogy, by contrast, connects elements of the past across time through transmission, interpretation, adaptation, and other kinds of preservation and change. Keywords Practice · Reconstruction · Genealogy · Culture · Context · Inscription · Tracing · Erasing Mathematics Subject Classification 01A85 (97A30 01-02 97-03)
1 Introduction: mathematics’ traces History is a tracing discipline. As much as pedagogues and researchers alike prize the value of imagining, illuminating, or recovering past worlds, the practice of history often hinges on other verbs—following, speculating, interpolating—that focus on what can be known indirectly from a past that is fundamentally unrecoverable and, in many respects, unimaginable. Historians make mediated understandings, deriving interpretations from materials available in the present that can be connected, one way or another, to the past. Historical events, contexts, and contingencies are both the
object of historical inquiry and that inquiry’s chief impediments, continually transforming what those in the past left behind. To learn and master history is to become a disciplined tracer, knowing how to reason from fleeting apparitions of past lives to coherent accounts of other times.1 Mathematica
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